Invariant totally geodesic unit vector fields on three-dimensional Lie groups

We give a complete list of left-invariant unit vector fields on three-dimensional Lie groups equipped with a left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group equipped with the Sasaki metric. As a result we obtain that each three-dimensional Lie group admits totally geodesic unit vector field under some conditions on structural constants. From a geometrical viewpoint, the field is either parallel or a characteristic vector field of a natural almost contact structure on the group.